Whakaoti i te 4x^2-16x+15

$$4 x ^ { 2 } - 16 x + 15$$

Tauwehe

$$\left(2x-5\right)\left(2x-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-16 ab=4\times 15=60$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 60.

$$-1,-60 -2,-30 -3,-20 -4,-15 -5,-12 -6,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

Ko te otinga te takirua ka hoatu i te tapeke -16.

$$a=-10 b=-6$$

Tuhia anō te 4x^{2}-16x+15 hei \left(4x^{2}-10x\right)+\left(-6x+15\right).

$$\left(4x^{2}-10x\right)+\left(-6x+15\right)$$

Tauwehea te 2x i te tuatahi me te -3 i te rōpū tuarua.

$$2x\left(2x-5\right)-3\left(2x-5\right)$$

Whakatauwehea atu te kīanga pātahi 2x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-5\right)\left(2x-3\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$4x^{2}-16x+15=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\times 15}}{2\times 4}$$

Pūrua -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 4\times 15}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-16\right)±\sqrt{256-16\times 15}}{2\times 4}$$

Whakareatia -16 ki te 15.

$$x=\frac{-\left(-16\right)±\sqrt{256-240}}{2\times 4}$$

Tāpiri 256 ki te -240.

$$x=\frac{-\left(-16\right)±\sqrt{16}}{2\times 4}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-16\right)±4}{2\times 4}$$

Ko te tauaro o -16 ko 16.

$$x=\frac{16±4}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{16±4}{8}$$

Nā, me whakaoti te whārite x=\frac{16±4}{8} ina he tāpiri te ±. Tāpiri 16 ki te 4.

$$x=\frac{20}{8}$$

Whakahekea te hautanga \frac{20}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{5}{2}$$

Nā, me whakaoti te whārite x=\frac{16±4}{8} ina he tango te ±. Tango 4 mai i 16.

$$x=\frac{12}{8}$$

Whakahekea te hautanga \frac{12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{3}{2}$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te \frac{5}{2} mō te x_{1} me te \frac{3}{2} mō te x_{2}.

$$4x^{2}-16x+15=4\left(x-\frac{5}{2}\right)\left(x-\frac{3}{2}\right)$$

Tango \frac{5}{2} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$4x^{2}-16x+15=4\times \frac{2x-5}{2}\left(x-\frac{3}{2}\right)$$

Tango \frac{3}{2} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$4x^{2}-16x+15=4\times \frac{2x-5}{2}\times \frac{2x-3}{2}$$

Whakareatia \frac{2x-5}{2} ki te \frac{2x-3}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$4x^{2}-16x+15=4\times \frac{\left(2x-5\right)\left(2x-3\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4x^{2}-16x+15=4\times \frac{\left(2x-5\right)\left(2x-3\right)}{4}$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4x^{2}-16x+15=\left(2x-5\right)\left(2x-3\right)$$

Aromātai

$$\left(2x-5\right)\left(2x-3\right)$$